Thanks, Allan. Am trying to clarify Cantor's contribution. Aristotle and Newton could sum infinite series, but if I've got it right they thought of infinity as a very large number and not something in a separate category. Am wondering how Zeno, Aristotle, Cauchy and Cantor would answer the following two questions. *1. Do these lines meet?* y = 1/2 + 1/4 + 1/8 ... Y = 2 2. Is an odometer displaying 0.999... indicating the same thing as an odometer displaying 1? - - - - I'm thinking the answers would be First question Zeno: No. y doesn't touch Y Aristotle: Yes. y touches Y eventually ("at infinity") but unclear how this happens. Cauchy: Yes, effectively. y touches Y effectively (can get as close as you want) Cantor: No. y will never touch Y. However: y', which is the infinite-x version of y, is identical to Y Second question: Zeno: no. Aristotle: you can make the display very, very long, until the display equals 1 Cauchy: you can make the gap between the odometer and 1 as close to zero as you want, so the odometer display essentially equals 1. Cantor: the display as is will never equal 1. You need a different design eg an odometer that circles around on itself. Once you have this, there is no gap between the display and 1. They indicate the same thing. On Thu, Nov 15, 2012 at 8:47 AM, Allan Wechsler <acwacw@gmail.com> wrote:
It seems odd to me to say that 0.999... = 1 is a "definition". I would certainly consider it a theorem. It can be proven in any axiomatization of the real numbers. The 19th century yielded the first clear constructions of the reals. People were summing infinite series well before Cantor.
On Thu, Nov 15, 2012 at 1:54 AM, Dan Asimov <dasimov@earthlink.net> wrote:
I think Cauchy laid to rest any questions about the concept of time being an aspect of the concept of convergence.
--Dan
On 2012-11-14, at 9:51 PM, Gary Antonick wrote:
unless I'm missing something, there would still be the following problems: Zeno's arrow will never actually hit the tree (eg 1/2 + 1/4 + 1/8 will never quite get to 1) a polygon with more and more sides will not quite ever become a circle 9/10 + 9/100 + 9/1000 etc will not quite ever equal 1
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